# Present Value of Ordinary Annuity

The ordinary annuity is an annuity, a stream of cash flows that occur after equal interval, in which each periodic cash flow occurs at the end of each period.

Many financial products are in fact annuities, for example bonds. Most bonds pay fixed coupon payments after equal interval from their issue date to their maturity date. Bonds are priced by discounting those coupon payments and the final terminal redemption value to time 0 based on the market interest rates.

## Formula

One way to find the present value of an ordinary annuity is to manually discount each cash flow in the stream using the formula for present value of a single sum and then summing all the component present values to find the present value of the annuity.

 Present Value = PMT (1 + r/m)(m×n)

Where PMT is the periodic payment in annuity, r is the annual percentage interest rate, n is the number of years between time 0 and the relevant payment date and m is the number of annuity payments per year.

Alternatively, we can calculate the present value of the ordinary annuity directly using the following formula:

 Present Value of Ordinary Annuity = PMT × 1 − (1 + r/m)(n×m) r/m

The same calculation can be conducted using Excel PV function. PV function syntax is PV(rate, nper, pmt, [fv], [type]). Where rate is the periodic interest rate (i.e. annual percentage rate divided by number of payments per year), nper is the total number of payments, pmt is the amount of payment, [fv] is an optional argument allowing us to specify if there is any final balloon payment. [type] is an optional argument that specifies whether the annuity is an ordinary annuity or an annuity due. By default, Excel assumes the annuity to be an ordinary annuity.

## Example

Let us use the present value of an annuity formulas to find price of treasury bond that has 2 years till maturity. The bond has a par value of \$100 and coupon rate of 3% thereby paying \$1.5 coupon after each six-month period. The market yield on the bond is 2.9%. Calculate the price of the bond.

The bond price equals the present value all bond cash flows, both coupon payment and the final redemption value. The coupon payments form an ordinary annuity because they are equal and occur after equal interval (i.e. 6 months) while the final redemption value i.e. \$100 paid back at the bond maturity date is a single sum.

We can manually calculate the bond price by individually discounting each coupon payment and the redemption value as follows:

Cash Flow Amount Present Value Factor Present Value
1st Coupon 1.5 0.9857 \$1.48
2nd Coupon 1.5 0.9716 \$1.46
3rd Coupon 1.5 0.9577 \$1.44
4th Coupon 1.5 0.9440 \$1.42
Maturity Value 100 0.9440 \$94.40
Total \$100.19

The present value factors are calculated using the formula for present value of a single sum of money.

The first four cash flows form an annuity and the final term is the present value of a single sum.

The present value of the coupon payments can be calculated as follows:

 Bond Price = Coupon × 1 − (1 + r/m)-(m×n) + FV r/m (1 + r/m)(m×n)
 Bond Price = \$1.5 × 1 − (1 + 2.9%/2)-(2×2) + \$100 = \$100.19 2.9%/2 (1 + 2.9%/2)(2×2)

If you want to calculate present value using Excel in the above situation, you need to enter the following function: PV(2.9%/2,4,-1.5,-100,0).